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Properties

Label 2-15-15.14-c10-0-4
Degree $2$
Conductor $15$
Sign $-0.109 - 0.993i$
Analytic cond. $9.53035$
Root an. cond. $3.08712$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.37·2-s + (224. − 92.2i)3-s − 1.01e3·4-s + (−862. + 3.00e3i)5-s + (758. − 311. i)6-s + 1.60e4i·7-s − 6.87e3·8-s + (4.20e4 − 4.14e4i)9-s + (−2.90e3 + 1.01e4i)10-s + 1.09e5i·11-s + (−2.27e5 + 9.33e4i)12-s + 5.72e5i·13-s + 5.40e4i·14-s + (8.31e4 + 7.54e5i)15-s + 1.01e6·16-s − 1.54e6·17-s + ⋯
L(s)  = 1  + 0.105·2-s + (0.925 − 0.379i)3-s − 0.988·4-s + (−0.275 + 0.961i)5-s + (0.0975 − 0.0400i)6-s + 0.952i·7-s − 0.209·8-s + (0.711 − 0.702i)9-s + (−0.0290 + 0.101i)10-s + 0.677i·11-s + (−0.914 + 0.375i)12-s + 1.54i·13-s + 0.100i·14-s + (0.109 + 0.993i)15-s + 0.966·16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 - 0.993i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.109 - 0.993i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ -0.109 - 0.993i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.997649 + 1.11358i\)
\(L(\frac12)\) \(\approx\) \(0.997649 + 1.11358i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-224. + 92.2i)T \)
5 \( 1 + (862. - 3.00e3i)T \)
good2 \( 1 - 3.37T + 1.02e3T^{2} \)
7 \( 1 - 1.60e4iT - 2.82e8T^{2} \)
11 \( 1 - 1.09e5iT - 2.59e10T^{2} \)
13 \( 1 - 5.72e5iT - 1.37e11T^{2} \)
17 \( 1 + 1.54e6T + 2.01e12T^{2} \)
19 \( 1 + 7.20e5T + 6.13e12T^{2} \)
23 \( 1 - 2.49e6T + 4.14e13T^{2} \)
29 \( 1 + 1.79e7iT - 4.20e14T^{2} \)
31 \( 1 - 1.83e7T + 8.19e14T^{2} \)
37 \( 1 - 6.73e7iT - 4.80e15T^{2} \)
41 \( 1 + 1.78e8iT - 1.34e16T^{2} \)
43 \( 1 - 1.66e8iT - 2.16e16T^{2} \)
47 \( 1 + 2.99e8T + 5.25e16T^{2} \)
53 \( 1 - 6.37e8T + 1.74e17T^{2} \)
59 \( 1 - 3.05e7iT - 5.11e17T^{2} \)
61 \( 1 - 9.80e8T + 7.13e17T^{2} \)
67 \( 1 - 6.91e8iT - 1.82e18T^{2} \)
71 \( 1 - 2.10e9iT - 3.25e18T^{2} \)
73 \( 1 - 2.47e9iT - 4.29e18T^{2} \)
79 \( 1 - 1.45e9T + 9.46e18T^{2} \)
83 \( 1 - 6.85e9T + 1.55e19T^{2} \)
89 \( 1 - 5.45e9iT - 3.11e19T^{2} \)
97 \( 1 + 3.04e9iT - 7.37e19T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.73121998246038070981624952013, −15.38306570220154526210673277131, −14.50590058906729366360187999634, −13.37649911071460522509525336595, −11.88420789742900123408006570630, −9.656650531312152709233519505471, −8.514902392639729922700059588824, −6.74128561105153195685037561125, −4.17981898079462158170595520228, −2.33082677815686450146134919917, 0.65720879401818681965842029171, 3.61668629100196574535833409449, 4.91866098434590377901557073339, 7.980052798506318295450052447452, 8.967136986716704179908168816287, 10.45999943399443166973074768056, 12.95389013035845155559694964346, 13.58486214786738384839080631992, 15.07013654193101881274029186979, 16.47954043874997401200186239965

Graph of the $Z$-function along the critical line